Toward Heisenberg scaling in non-Hermitian metrology at the quantum regime

Non-Hermitian quantum metrology, an emerging field at the intersection of quantum estimation and non-Hermitian physics, holds promise for revolutionizing precision measurement. Here, we present a comprehensive investigation of non-Hermitian quantum parameter estimation in the quantum regime, with a special focus on achieving Heisenberg scaling. We introduce a concise expression for the quantum Fisher information (QFI) that applies to general non-Hermitian Hamiltonians, enabling the analysis of estimation precision in these systems. Our findings unveil the remarkable potential of non-Hermitian systems to attain the Heisenberg scaling of 1/t, where t represents time. Moreover, we derive optimal measurement conditions based on the proposed QFI expression, demonstrating the attainment of the quantum Cramér-Rao bound. By constructing non-unitary evolutions governed by two non-Hermitian Hamiltonians, one with parity-time symmetry and the other without specific symmetries, we experimentally validate our theoretical analysis. The experimental results affirm the realization of Heisenberg scaling in estimation precision, marking a substantial milestone in non-Hermitian quantum metrology.


I. THEORY
A. Detailed derivation of QFI for general non-Hermitian Hamiltonians Consider a non-Hermitian Hamiltonian Ĥα with unknown parameter α, the evolution operator can be written as Ûα = e −i Ĥαt (40), and the probe state is prepared to be ρ 0 = |ψ 0 ψ 0 |.
In non-Hermitian systems, the evolution operator U α is non-unitary, so the output state will no longer be normalized after evolution.However, as for measurement process, the probabilities of measurement outcomes sum up to 1. Therefore, the output state can be written as ( 64) . Then, we rewrite the QFI and associate it with h α .Let K α = Tr{ρ α }, then we have |ϕ α = U ˆα|ψ 0 / √ K α , the first term ∂ α ϕ α |∂ α ϕ α can be expand as follow Similarly, the second term | ∂ α ϕ α |ϕ α | 2 can be written as Collecting this two results, we obtain (S6) As we discussed above, ĥ = i(∂ α Ûα ) Û −1 α , then F α can further written as F α = 4( ĥ † ĥ α − ĥ † α ĥ α ), where If the Hamiltonian is multiplicative H ˆ = Gθ, we have U ˆ = e −iGθt = e −iGα , where θ is the parameter which we aim to estimate and α = θt.If the evolution time t is a constant then estimating α is equivalent to estimate θ, Eq. (S7) return the form Clearly, the key to QFI is the local generator ĥ of the parametric translation of Ûα .In general, it is difficult to derive ĥ, one will find that ∂ α Ĥα does not commute with Ûα when calculate ∂ α Ûα .Fortunately, this problem has been discussed in detail already (80), the generator ĥ can be expressed as ( 61) Notably, QFI represents the ultimate precision for a single measurement.However, in non-Hermitian systems, there is gains and losses during the non-unitary evolutions.Relying solely on QFI is insufficient to characterize the ultimate estimation precision when considering fixed resources of probe states.Considering the effect of gains and losses on precision, we multiply the QFI by the normalization coefficient K α of the output state This quantity actually characterized the estimation precision for evolution Ûα in the case of fixed resources of probe states.
Here, we further discuss the precision scaling of a PT -symmetric system ĤPT which is embedded into a larger Hermitian system (62,75).It is well-known that some PT -symmetric Hamiltonians are pseudo-Hermitian, we can find an invertible Hermitian operator η that satisfies η ĤPT = Ĥ † P T η.Here, we define c := N i=1 1/λ i and ζ = cη − Î, where λ i represents the ith eigenvalue of η and Î is the identity operator.We consider the case of PT -symmetry unbroken, so η and ζ are positive.The two-qubit dilation Hermitian system can be written as (62,75) where Ĥs and V are Hermitian and satisfy Solving the above equation, we obtain An entangled state |Ψ tot (t) in two-qubit dilation Hermitian system can be written as where |ψ P T (t) = e −i ĤP T t |ψ 0 is the state evolves in PT -symmetric system and |ψ 0 is the initial state of |ψ P T (t) .It can be proved that |Ψ tot (t) satisfies the Schördinger equation of Ĥtot , we have where a and b are complex numbers, then we define the variance of non-Hermitian operator Ô as (∆ Ô) 2 = ψ| Ô † Ô|ψ − ψ| Ô † |ψ ψ| Ô|ψ .We can further rewrite it as follow We can see that for two-level systems, the projective operator P = I − |ψ ψ| = |φ φ| is exactly the pure state that orthogonal to |ψ .Assume |φ = c|λ 1 + d|λ 2 , then we have and Ô|ψ can be written as Ô|ψ = aλ 1 |λ 1 + bλ 2 |λ 2 .Therefore, the variance can be further written as (S19) According to Eq. (S18), we have Obviously, for a given state |ψ = a|λ 1 + b|λ 2 , the variance (∆ Ô) 2 is determined by the modulus of difference between two eigenvalues λ 1 and λ 2 .As we discussed in section A, for a multiplicative pseudo-Hermitian Hamiltonian Ĥ = Gs, the QFI is the variance of the G which has the same properties with Ĥ.It is well-known that for pseudo-Hermitian Hamiltonians, the eigenvalues gradually degenerate as EP is approached.Therefore, for a given state |ψ = a|λ 1 + b|λ 2 , the QFI of multiplicative pseudo-Hermitian Hamiltonian Ĥ would tends to zero near EP.
However, as for non-multiplicative Hamiltonians, QFI is the variance of the generator h, the properties of h near EP could be much different from that of Hamiltonian.For example, consider a non-Hermitian Hamiltonian Ĥ1 = I sin ασ z + cos ασ x , α is the unknown parameter, the evolution operator is U 1 = e −iH 1 t .Then we can obtain the eigenvalues of local generator h 1 increases as EP (α = π/4) is approached.Thus, the QFI of non-multiplicative Hamiltonians could increase near EP.Noticed that the above discussion applies to the case that EP is not exactly reached.Because two eigenstates would coalesce at EP, the state |ψ = a|λ 1 + b|λ 2 cannot be able to represent an arbitrary state anymore, it is just the eigenstate.
Furthermore, we are also able to analyse that whether we can achieve Heisenberg precision for multiplicative non-Hermitian Hamiltonians.For example, the generator of the parameter α for general time t of the non-Hermitian Hamiltonian ĤPT (α) in the main text is and we can obtain the eigenvalues of ĥα (t) as follows As shown in Fig. S1, the growth of the modulus of difference |λ + − λ − | reaches the scale of t.According to Eq. (S20), we can derive that the growth of of QFI reaches t 2 , i.e., Heisenburg limit, and we obtain the exact QFI for general time t when probe state is |ψ 0 = |0 as follows Obviously, there is t 2 in F α (t).

C. Proof of the condition for optimal measurements
Based on the expression of QFI we proposed, we give the detailed proof of the condition for optimal measurement in this section.Consider an arbitrary Hermitian operator Â as measurement operator.According to the error-propagation formula (65,66), we have where |ϕ α is the normalized output state after evolution.For simplicity, let where Substitute this result into Eq.(S27), we obtain Here, we define an operator δ M = M − M α , the variance of M over |ϕ α is defined by According to the non-Hermitian uncertainty relationship (67-71), we have Obviously, the expression in the last brackets of Eq. (S29) is the same as the result in Eq. (S31), so we have Define |f = δ ĥ|ϕ α and |g = δ Â|ϕ α , we have f |g = ĥ † Â α − ĥ † α Â α .Then, we further obtain the inequality as follow According to Cauchy-Schwarz inequality, the first inequality is saturated when |f and |g is linearly related, i.e., |f = c|g .And the second inequality is further saturated if c is an imaginary number.Therefore, we could derive the inequality it is exactly the QCRB, as we discussed above, it is saturated when where c is a real number.

D. The reason for normalization
It is well-known that the probability may not conserve in non-Hermitian systems evolution.However, for measurement process, the probabilities of measurement outcomes always sum up to 1, even if there are losses or gains during evolution process, and the final state is normalized by dividing its trace.As shown in Fig. S2, we use the polarization of photons to encode states, where horizontally polarized state |H is encoded as |0 , and vertically polarized state |V is encoded as |1 , the photons are prepared as initial state ρ 0 = |ψ 0 ψ 0 | and then evolves in the non-Hermitian quantum system, which will lead to the decay of the total number of photons N tot , i.e., N = N tot Tr{ρ θ } < N tot .However, the probability is still normalized when we measure the final state ρ θ = U θ ρ 0 U † θ .For instance, we measure the final state with the set of projection operation Π = i |i i|, the probabilities of two measurement outcomes respectively are P 0 = N H /N = (N tot Tr{ρ θ |0 0|})/(N tot Tr{ρ θ }) and where N H and N V respectively are the number of horizontally polarized photons and vertically polarized photons.For measurement, the final state is actually Therefore, we normalize the final state by dividing its trace, for pure states, the normalized final state is

II. EXPERIMENT
Before we introduce our experimental setup, we first illustrate the optics elements in our experiment.
1) The Jones matrix of half-wave plate (HWP) and quarter-wave plates (QWP) in our experiment respectively are where θ is the angle between the light polarization direction and the fast axis of the wave plate.
2) The polarization beam splitter (PBS) in our experiment transmits the horizontal polarized photons and reflect the vertical polarized photons, as shown in Fig. S3A.
3) The beam displacer (BD) in our experiment transmits the horizontal polarized photons, but separates the vertical photons into new path which is about 4mm from the original path, as shown in Fig. S3B.

A. Photon-pair source
The central wavelength of pump laser in our experiment is 405 nm.The energy of pump laser is adjust by QWP1, HWP1 and PBS1, meanwhile the state is purified into |H .If the initial state is prepared as In our experiment, only the heralded single photon source is needed, so we prepare the initial state as |H .A dichroic mirror (DM) reflect pump light into the triangle sagnac interferometer, and then PPKTP crystal is clockwise pumped.After type-II phase-matched spontaneous parametric down-conversion process, one 405 nm photon splits into two 810 nm photons (|H → |H 1 |V 2 ).The DPBS and two mirrors are affective for both 405 nm and 810 nm photons, but DM does not reflect 810 nm photons.So these two 810nm photons are separated into two paths, then filtered by long-wave path filter (LPF) and collected into single-mode fibers.In our experiment, the power of pumped light is 2 mw, and we could obtain 80000 coincidences per second.

B. Theoretical framework of ÛPT evolution
The non-unitary PT -symmetric evolution ÛPT consists of the probe qubit (|H, V ) and ancilla qubit (|a, b ), by performing a projective operator on ancilla qubit, we could effectively construct a non-unitary PT -symmetric evolution Û P T = F ÛPT (26), as shown in Fig. S5.The non-unitary PT -symmetric evolution ÛPT can be written as The operator Ûtot is unitary, but in Fig. S5, we do not concentrate on the photons (a new qubit) losses at H3, H4 and PBS2, the left two-qubit evolution Û can be expressed as follow, where p = sin 2(φ 1 − φ 2 ) and q = cos 2φ 2 are controlled by H3 (φ 1 ) and H4 (φ 2 ), Ûtot is an unitary evolution.As shown in Fig. S6, p and q are controlled by H3, H4 and PBS2, |ϕ H and |ϕ V are prepared by H5, Q1, H6 and Q1 respectively, |ϕ ⊥ H and |ϕ ⊥ V are the corresponding orthogonal states.After preparing |ϕ H |a and |ϕ V |b , BD2 and BD3 combine their horizontal and vertical components into two paths respectively.Then the projective operator is realized by PBS3 and H9 set at 22.5 • , it can be expressed as where I is identity operator.The evolution after performing projective operator is In our experiment, the initial state of ancilla qubit is always |a , then, for an arbitrary initial state |ψ 0 = cos 2φ|H + sin 2φ|V of probe qubit, the final state after evolution is (p cos 2φ|ϕ H + q sin 2φ|ϕ V )/ √ 2. Thus, the evolution we construct is proportional to the theoretical non-Hermitian system evolution, for probe qubit, we have where F = γ/ √ 2 is a scalar function.

C. Optimal measurements
In this section, we prove that Â = |0 0| is optimal measurement for both multiplicative and non-multiplicative Hamiltonians when the probe state is |0 .In the case of estimate s, according to the condition for optimal measurements.
so the measurement Â is indeed the optimal measurement when estimate s.
In the case of estimate α, we further experimentally verify the result.We first give the theoretical analysis.the probe state is prepared as an arbitrary linear polarization pure state |ψ 0 = cos 2φ|0 + sin 2φ|1 , we change the probe state from |0 to |1 (φ = 0 • ∼ 45 • ) and perform Â = |0 0| as measurement.According to Eq. (S35), we have we can see that the condition for optimal measurement is satisfied only if φ = kπ/4, i.e., the probe state is |0 or |1 .
In our experiment, we set that s = 1, α = π/10 and t = π/[2s cos(π/10)].To get the statistic of the estimation, we make 5000 maximum likelihood estimates and get the distribution of estimators α, the experimental results is shown in Fig. S7.Due to the fluctuation of the data, the estimator α cannot be calculated by maximum likelihood estimation when φ = 22.5 • , φ = 27 • and φ = 31.5• , so the corresponding deviations are not plotted.As shown in Fig. S7B, the standard deviation of estimator reaches QCRB when the probe state is |0 and |1 , it is consistent with our theoretical analysis.We also give the exact data of Fig. S7, as shown in

D. Maximum likelihood estimation
When we estimate the parameter with different probe states, we apply maximum likelihood estimation to get the estimator α.For two-level systems, the distribution of measurement outcomes obeys binomial distribution, for n independent measurements, the likelihood function is this expression represents the probability that there are x measurement outcomes |0 in n measurements, and p(α) = | 0|ϕ α | 2 is the probability of obtaining |0 .The logarithmic of likelihood function is Then we solve the differential equation The solution of the equation is the estimator α = α(n, x), it is the function of n and x.Substituting the experimental results into α(n, x), we obtain the estimate of the parameter α.And we further obtain the distribution of estimation by repeating this process.

E. Experimental datas of reaching Heisenberg scaling
In the main text, we show the standard deviation of estimation results and a part of distribution of estimators, here we present more detailed results.As shown in Table .S3 and Table.S4, we give the exact experimental and theoretical results for different time points.In Table .S3 and Table.S4, we can see that the average of estimator ŝ and α is slightly different from the theoretical value s = 1 and α = π/4, because of the error of the non-unitary PT -symmetric evolution Û P T we constructed.As show in Fig. S8 and Table.S5, we can see that although the errors of the probabilities we measured is quite small, there are still errors between the average of estimators and theoretical value, especially for the probe states that lead to worse estimation precision.

F. Non-Hermitian Hamiltonian without PT or anti-PT symmetry
To demonstrate the generality of our theory, we consider a Hamiltonian without any special symmetries given by where κ is the unknown real parameter and κ = 1.It can be seen that Ĥκ is neither PT -symmetric nor anti PT -symmetric.The corresponding evolution operator Ûκ can be deduced from the Hamiltonian Ĥκ as According to Eq. (S2), we can obtain the generator as The eigenvalues of ĥκ (t) are λ ± = ± [−1 + 2κt 2 + cos(2t √ κ)]/(8κ).As shown in Fig. S9, the growth of the modulus of the difference, |∆λ| = |λ + − λ − |, reaches the scale of t, indicating that the growth of the quantum Fisher information scales as t 2 .The initial probe state is prepared as |ψ 0 = |0 .By utilizing Eq. (S7), we can calculate the QFI as It can be observed that the QFI indeed exhibits a growth scaling of t 2 , which represents the Heisenberg scaling.
Using a similar scheme and experimental setup in Fig. S5 and S6, we realize the non-unitary evolution operator Û κ = F Ûκ , where F is a real scalar.It is important to note that the states |ϕ H,V realized in path a and b differ from those in the experiment of PT -symmetry Hamiltonian, despite employing a similar experimental scheme and setup.In the previous experiment, the states in two paths are In this experiment, the total evolution operator is given by where By performing a projective operator P = I ⊗ (|a + |b )( a| + b|)/2, we obtain non-unitary evolution operator on probe qubit as follows where |ϕ H and |ϕ V are controlled by the angle of H5, Q1, H6 and Q2.For each time point, we adjust these wave plates to ensure that the evolution operator is Û κ (t).To illustrate the distinctions between the two experiments, we present the angle configurations of H6 and Q2 at the same time points in both experiments (here, the probe state is |0 , and it does not pass through H5 and Q1), as shown in Table .S6.
We input the probe state |ψ 0 = |0 and perform n = 1000 ∼ 1200 measurements Â = |0 0| on output states for varying time t, where Â is the corresponding optimal measurement for |ψ 0 according to the condition for optimal measurements Eq. (S35), The actual value of parameter is κ = 2.For each time point, we repeat maximum likelihood estimation for 1000 times to obtain the distributions of the estimator κ.As shown in Fig. S10 , we present the estimation precision for varying time.The experimental results reveal that the precision follows Heisenberg scaling t which aligns with the theoretical prediction.The exact experimental data is shown in Table .S7.For this Hamiltonian without specific symmetries, the experimental results also match with the theory, which exhibits the generality of our theory.

Figure S6 :
Figure S6: Schematic illustration of the non-Hermitian system evolution.

Figure S7 :Figure S8 :Figure S9 :
Figure S7: Estimation precision.(A) The probabilities of measurement outcomes for different probe states after evolution.We set the measurement as Â = |0 0|, the probe states |ψ 0 = cos 2φ|0 + sin 2φ|1 is changed from φ = 0 • to φ = 45 • .(B) The standard deviation.The experimental standard deviation (black dots) approximately matches well the theoretical estimation precision (orange solid lines) calculated by the error-propagation formula.The ultimate precision described by QFI (blue dashed lines) is achieved when the probe state is |0 or |1 .

Table S1 :
FigureS10: QFI for varying time t.The probe state is set as |0 , the measurement performed is Â, and the condition for optimal measurements is satisfied.The practical value of κ we set is 2. (A) The square root of QFI, the orange dots are the experimental data and the orange solid line is the theoretical value of F κ (t).(B) The QFI multiplied by normalized coefficient K κ .The measurement results for different probe states.

Table S2 :
The standard deviations of α for different probe states.

Table S3 :
The standard deviations of ŝ for different times.

Table S4 :
The standard deviations of α for different times.

Table S5 :
The measurement results for different evolution times.

Table S7 :
The standard deviations of κ for different times.